Pappus’s theorem: A number of theorems attributed to Pappus of Alexandria including the Pappus's Centroid Theorems which intuitively states that the area of a surface of revolution can be calculated by finding the product of an arc length and the length travelled by its centroid and the volume of a revolution definied by a curve f can similarly be found by finding the product of the area under f and the length travelled by its centroid.

An alternative definition defines the locus of all points that is equidistant from a point known as the focus, and a line known as the directrix. A parabola is specified as long as the focus does not lie on the directrix.

All parabolae are effectively of the same shape (with eccentricity 1), being scaled versions of other parabolae. Also, all parabolae are symmetirc about the normal of the apex (vertex).

paradox: The situation where valid inferences made on valid premises seemed to imply a contradiction. Note that the mathematical (logical) definition of a paradox is much narrower than in everyday use.

parallel: Describing lines (or other geomtric objects) that are non-intersecting, "going" in the same direction and keep equal distance everywhere. For lines, these three concepts are exactly the same in Euclidean geometry, while in other geometries, the concept of parallel (without further clarifications) can be taken to mean an extension to any of these, given that this concept on geometric objects originated from related concepts that happen to be the same in Euclidean geometry.

parallel postulate: An assertion by Euclid in his book Elements. It was presented as a postulate (after 4 others) without being proven. It effectively states that two non-parallel lines must meet.

This "postulate" has resisted proof for many centuries before consideration is given to the possibility that it is simply not necessarily true. This leads to the development of non-Euclidean geometry, while the familiar geometry in which the fifth postulate is true is known as Euclidean Geometry. It should be noted that we now know of the parallel postulate's independence from the other postulates, that is, the parallel postulate cannot be proven from the four other postulates. In that sense, the parallel postulate of Euclid is more of an axiom for a particular geometric system.

parametric equations: A way of describing relationships between variables through other variables known as parameters. For example, we may define the relationship between two variables x and y through a variable t, which we are not interested in. (i.e. we would not particularly want to plot a graph with x, y and t axis, but rather x and y axis only. The variable t is only present to facilitate the description of the relationship between x and y)

Some equations would be parametric simply because it is not easy (or is impossible) to be stated in ordinary (i.e. explicit or implicit) form. For example, x = 2t + t2 and y = tt should be difficult to express without the parameter t or any other "intermediaries".

parent functions: A collection of simple functions used to build more complicated functions. In this sense there are no definitions of what constitue "simple" and is even more vague than the idea of elementary functions.

parentheses: Pairs of symbols ( ) used to indicate the priority in calculations that may break away from conventions (e.g. left to right, multiplications before additions), or provides information into the preferred grouping of multiple objects. (e.g. a x b x c for the vector triple product) Often known as brackets or round brackets.

Usually the number of terms is specified and we must sum the first specified nuber of terms without omitting any. The concept of partial sums allow the definition of the convergence of a series to be based on that of a sequence. Examining the partial sums of a series also helps evaluate the sum of infinite sereis.

2. The number of ways of choosing r elements from a set of n elements when the order of being chosen matter. This is the same as performing (the equivalent actions associated with) combination followed by the factorial.

phase: The position within a cycle for a periodic system. Due to the nature of periodic systems there is usually no "natural" starting point of a cycle and one would be needed as a reference point most of the time to quantify the phase.

pinching theorem: Also known as the Squeeze theorem. A theorem that states that, if a function is known to be less than or equal to (or alternatively, more than or equal to) a function, then its limit shall also be less than or equal to (or correspondingly, more than or equal to) the limit of the bounding function. In one application, finding two functions (one bounding above and one below) with the same limit forces the squeezed function to have the same limit as the other two.

pint: An British Imperial unit of volume equal to an eighth of a (British Imperial) gallon or 20 (Bitish Imperial) fluid ounces.

planar graph: In graph theory, a graph that can be represented on a plane without any intersection of its edges. Note that a planar graph is an inherent property of the graph invariant to its representation while a plane graph refers to the representation in a plane.

plane: A set of points which can be defined by (but not uniquely) the location of a point (on the plane), and two directions (parallel to the plane but not parallel to each other).

It is also seen in compound angle formulae in trigonometry for the convenient expression of pairs of related identities, where the minus/plus sign is also seen. In this case, the minus/plus sign serves the same function but correspond to the opposite cases. (i.e. When the "top" case is chosen, we use the addition in the plus/minus sign and the subtraction in the minus/plus sign, and when the "bottom" case is chosen, we use the subtraction in the plus/minus sign and the addition in the minus/plus sign.)

Also note that another common use of the sign to indicate a range (such as a confidence interval is not a strict mathematical use. 45% ± 3%.

A polygon can be classified as equilateral (all sides are of the same length), equiangular (all angles are the same), regular (both equiangular and equiangular) or none of the above. It can also be independently classified into convex (all interior angles are less tha 180°) or concave (at least one angle is reflex - more than 180°).

The terms are identified by the power that the variable is raised to and the corresponding constant in the term is known as a coefficient of that term.

The leading coefficient of a polynomial is the coefficient of the term of highest power in a single variable case. (It is "leading" due to the convention of writing finite number of terms in descending order, this leading coefficient often determines the golbal behaviour of the polynomial.)

Polynomials of lower orders have special names: a polynomial of highest power of 1 is linear, 2 is quadratic, 3 is cubic, 4 is quartic and 5 is quintic. Special names are seldom used in powers higher than 5.

positional system: A system representing numbers where symbols can take on different (but usually related) meanings depending on its (relative) position within the collection of symbols.

position vector: A vector used to represent the position of a point via the relative position of this point to a known reference universal within the coordinate system, such as the origin0. Often denoted by the symbol r.

positive number: Any number greater than zero. Note that this definition would only apply if such a comparison could be made. (For example, while one of the three statements a < b , a = b , a > b must be true for real numbers, it is not the case for complex numbers, thus the concept of "greater than" and by extension "positive" may not make sense.)

Note also that the set of positive numbers is slightly different from the set of non-negative numbers. (Mainly because even a real number can be neither positive or negative.)

prefix notation: Also known as Polish notations, where the operator is expressed before the operands. Such a system has as its advantage simpler structure requiring very few brackets (or none at all), but may cause confusion unless other ways are used to indicate multiplication, such as 2x to mean the product of 2 and x, and for the positional system of representing numbers where 34 means the sum of thirty and four.

pressure: The measure of force exerted over a standardised area. Often denoted with the Greek letter p (rho). The S.I. unit of pressure is the Pascal (named after Blaise Pascal), being the equivalent of one Newton with a square metre.

prime (prime number): A positive integer with exactly 2 factors. (Where 1 is necessarily one of the factors and itself another.) While 1 has been considered a prime number at times it is usually not by the aforementioned modern deion.

prime symbol: Also known as the accent or simply "dash". Usually placd on the top right of other mathematical symbols to indicate derivatives such as f'(x), related objects such as point p and p' or arbitrary constant C and C', measurements such as feet and minutes (angles).

probability paper: A type of graph paper where the scales on the axes are manipulated such that a certain distribution represents a straight line on the paper such that it makes for easier fitting of data to the specified distribution.

product rule: A theorem in differentiation that provides an easy way to differentiate the product of two functions. It can be easily proven from first principles and can be easily extended to cases covering the product of three or more functions. It is also easy enough to remember for most without resorting to commiting the following symbols to memory.

2. A sequence where each term is determined by a fixed (simple) rule and the preceeding term.

projectile: An object purturbed from stationary by a force before being subjected to air resistance and gravity only - where the effects of air resistance is generally understood to not reduce the speed of the object.

proof: A sequence of finite number of statements, each of which is either an axiom or the result from rules of inference on statements that appeared before.

proof by contradiction: A method of proof based on the fact that a logical sentence and its contrapositive always hold the same truth value. One attempts to prove a sentence by contradiction by assuming the inverse of the desired statement and shows that it must lead to some logical contradiction. Thus, "if the negation of the original statement leads to a contradiction means no contradiction arises leads to the original statment being true.".

proper divisor: A non-trivial factor (i.e. divisor). The trivial factors are 1 and the number itself, since they are always factors regardless of the number being divided.

proper fraction: A fraction whose numerator is strictly less than the denominator. The same concept can be extended to cover algebraic (rational ) fractions by insisting that the order of the polynomial in the numerator be less than the order of the polynomial in the denominator.

proper subset: A proper subset is one completely contained within another. X is a proper subset of Y is al elements of X are elements of Y (this qualifies X as a subset of Y), and there is at least one element of Y that is not in X. (This qualifies for the "proper")

p-value: The probability that a test statistic take the value of, or more deviated than, the actual value observed. Intuitively speaking this calculates the probability that the observed result happens by chance rather than other reasons. We reject the null hypothesis if the p-value of an observed value of a test statistic is less than that of a pre-determined significance level.